Solve 7-sqrt{x}=sqrt{x-21}

$$7 - \sqrt { x } = \sqrt { x - 21 }$$

Solve for x

$$x=25$$

Solution Steps

Square both sides of the equation.

$$\left(7-\sqrt{x}\right)^{2}=\left(\sqrt{x-21}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-\sqrt{x}\right)^{2}.

$$49-14\sqrt{x}+\left(\sqrt{x}\right)^{2}=\left(\sqrt{x-21}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$49-14\sqrt{x}+x=\left(\sqrt{x-21}\right)^{2}$$

Calculate \sqrt{x-21} to the power of 2 and get x-21.

$$49-14\sqrt{x}+x=x-21$$

Subtract x from both sides.

$$49-14\sqrt{x}+x-x=-21$$

Combine x and -x to get 0.

$$49-14\sqrt{x}=-21$$

Subtract 49 from both sides.

$$-14\sqrt{x}=-21-49$$

Subtract 49 from -21 to get -70.

$$-14\sqrt{x}=-70$$

Divide both sides by -14.

$$\sqrt{x}=\frac{-70}{-14}$$

Divide -70 by -14 to get 5.

$$\sqrt{x}=5$$

Square both sides of the equation.

$$x=25$$

Substitute 25 for x in the equation 7-\sqrt{x}=\sqrt{x-21}.

$$7-\sqrt{25}=\sqrt{25-21}$$

Simplify. The value x=25 satisfies the equation.

$$2=2$$

Equation -\sqrt{x}+7=\sqrt{x-21} has a unique solution.

$$x=25$$